Consider an elastic string under tension which is at rest along the
dimension. Let
,
, and
denote the unit vectors in
the
,
, and
directions, respectively. When a wave is
present, a point
originally at
along the string is
displaced to some point
specified by the displacement
vector

Note that typical derivations of the wave equation consider only the
displacement
in the
direction. This more general treatment
is adapted from [122]. An alternative clear
development is given in [394].

The displacement of a neighboring point originally at
along the string can be specified as

Let
denote string tension along
when the string is at rest, and
denote the vector tension at the point
in the present displaced
scenario under analysis. The net vector force acting on the infinitesimal
string element between points
and
is given by the vector sum of
the force
at
and the force
at
, that is,
. If the string
has stiffness, the two forces will in general not be tangent to the string
at these points. The mass of the infinitesimal string element is
,
where
denotes the mass per unit length of the string at rest. Applying
Newton's second law gives

(B.31)

where
has been canceled on both sides of the equation. Note
that no approximations have been made so far.

The next step is to express the force
in terms of the tension
of the string at rest, the elastic constant of the string, and
geometrical factors. The displaced string element
is the
vector